Derivatives can be analyzed in functions and further extend to the concepts in limits. The general trend shows indirect questions are asked from this topic in JEE Main and JEE Advanced. However, in BITSAT and state engineering entrances exams, direct questions may be asked.
1. Increasing and Decreasing functions
If a function is increasing or decreasing in the entire domain of the function is called a monotonic function.
If a function is either increasing or decreasing in its domain, the function is called as a non-monotonic function.
A function is said to be increasing at a point x=a in the function, if
A function is said to be decreasing at a point x=a in the function, if
2. Point of inflection
Point of inflection is the point in the curve where the second derivative of the function changes its sign or in other words, the inflection point is the point where the second derivative is zero.
The term 'extremum ' is applicable for maximum value as well as the minimum value.
The critical point of a function is the point where either the function is not differentiable or where the derivative of the function is zero.
The critical points are also called stationary points because there is no change in the value of the function at this point.
4. The concept of minima and maxima
Local maxima: A point on the curve where the value of the function at the point is more than the limiting value of the function.
Local minima: A point on the curve where the value of the function at the point is less than the limiting value of the function.
Global maxima: The maximum value of the function among the different critical points of the function.
Global minima: The minimum value of the function among the different critical points of the function.
5. Test of the derivative
The point of minima is a point on the curve of the function, where the first derivative is zero and the second derivative is positive.
The point of maxima is a point on the curve of the function, where the first derivative is zero and the second derivative is negative.
In case the second derivative is zero, we continue the test on higher order derivatives until a non-zero derivative occurs. Then check the positive/negative nature of the nth derivative.
6. Points to remember
(a) A function may have several local maxima and local minima but will only have one global maximum and one global minimum.
(b) A local maximum/minimum may or may not be the global maxima/global minima.
(c) A local maximum may be less than a local minimum at some other point.
(d) For any continuous function, the occurrence of minima and maxima is alternate.
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